Science:Math Exam Resources/Courses/MATH100/December 2010/Question 01 (k)
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Question 01 (k) 

If and for , what is the least could possibly be? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Let's look at all the information we have. We know that we are looking at a function on an interval where we know something about its derivative and about the value of the function at 0. Do we have a formula or a theorem that can relate all these values? 
Hint 2 

Try the mean value theorem. 
Hint 3 

The mean value theorem states the following: If is continuous on and differentiable on , then there exists a point such that . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Notice that is differentiable on and hence it is continuous there. Thus, by the mean value theorem, there exists a point such that . We know that on that . Since and , we have that . Solving gives and hence the minimum could be is 22. 